Unconstrained optimization. The asymptotic rate of convergence and asymptotic normality have been established for the averaged stochastic gradient descent (ASGD) [1]}. Subsequently, results on the asymptotic normality of other first-order methods and on the construction of the covariance estimator have been reported [2]}, [3]}, [4]}. The vast majority of the existing works on the asymptotic analysis of SGD focused on the averaged iterate, and excluded the stepsize \(1/t\)  due to some technical challenges. Recently, the asymptotic analysis of stochastic Newton methods has been proposed. [5]} designed a Newton scheme for logistic objective, and [6]} extended to general strongly convex objectives. Compared to the literature on first-order methods, both works showed the normality of the last iterate with \(1/t\)  stepsize. However, those analyses are not applicable for Problem (REF ) due to the following reasons.
